Functional neuroimaging data embodies a massive multiple testing problem, where 100 000 correlated test statistics must be assessed. The familywise error rate, the chance of any false positives is the standard measure of Type I errors in multiple testing. In this paper we review and evaluate three approaches to thresholding images of test statistics: Bonferroni, random �eld and the permutation test. Owing to recent developments, improved Bonferroni procedures, such as Hochberg’s methods, are now applicable to dependent data. Continuous random �eld methods use the smoothness of the image to adapt to the severity of the multiple testing problem. Also, increased computing power has made both permutation and bootstrap methods applicable to functional neuroimaging. We evaluate these approaches on t images using simulations and a collection of real datasets. We �nd that Bonferroni-related tests offer little improvement over Bonferroni, while the permutation method offers substantial improvement over the random �eld method for low smoothness and low degrees of freedom. We also show the limitations of trying to �nd an equivalent number of independent tests for an image of correlated test statistics. 1

We consider identifying differentially expressing genes between two patient groups using microarray experiment. We propose a sample size calculation method for a specified number of true rejections while controlling the false discovery rate at a de-sired level. Input parameters for the sample size calculation include the allocation proportion in each group, the number of genes in each array, the number of differen-tially expressing genes, and the effect sizes among the differentially expressing genes. We have a closed-form sample size formula if the projected effect sizes are equal among differentially expressing genes. Otherwise, our method requires a numerical method to solve an equation. Simulation studies are conducted to show that the calculated sample sizes are accurate in practical settings. The proposed method is demonstrated with a real study. Key words: Block compound symmetry, Family-wise error rate, Prognostic gene, True rejection, Two-sample t-test.

In structural brain MRI, group differences or changes in brain structures can be detected using Tensor-Based Morphometry (TBM). This method consists of two steps: (1) a non-linear registration step, that aligns all of the images to a common template, and (2) a subsequent statistical analysis. The numerous registration methods that have recently been developed differ in their detection sensitivity when used for TBM, and detection power is paramount in epidemological studies or drug trials. We therefore developed a new fluid registration method s the mappings and performs statistics on them in a consistent way, providing a bridge between TBM registration and statistics. We used the Log-Euclidean framework to define a new regularizer that is a fluid extension of the Riemannian elasticity, which assures diffeomorphic transformations. This regularizer constrains the symmetrized Jacobian matrix, also called the deformation (or strain) tensor. We applied our method to an MRI dataset from 40 fraternal and identical twins, to revealed voxelwise measures of average volumetric differences in brain structure for subjects with different degrees of genetic resemblance.

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Some extended false discovery rate (FDR) controlling multiple testing procedures rely heavily on empirical estimates of the FDR constructed from gene expression data. Such estimates are also used as performance indicators when comparing different methods for microarray data analysis. The present communication shows that the variance of the proposed estimators may be intolerably high, the correlation structure of microarray data being the main cause of their instability. 1.

Abstract. Tensor-based morphometry (TBM) is an analysis approach that can be applied to structural brain MRI scans to detect group differences or changes in brain structure. TBM uses nonlinear image registration to align a set of images to a common template or atlas. Detection sensitivity is crucial for clinical applications such as drug trials, but few studies have examined how the choice of deformation model (regularizer or Bayesian prior) affects sensitivity. Here we tested a new registration algorithm based on a fluid extension of Riemannian Elasticity [17], which penalizes deviations from zero strain in a log-Euclidean tensor framework, but has the desirable property of enforcing one-to-one mappings. We compared it to a standard large-deformation continuummechanical registration approach based on hyperelasticity. To compare the sensitivity of the two models, we studied corpus callosum morphology in 26 HIV/AIDS patients and 12 matched healthy controls. We analyzed the spatial gradients of the deformation fields in a multivariate Log-Euclidean framework [1] [12] to map the profile of systematic group differences. In cumulative p-value plots, the Riemannian prior detected disease-related atrophy with greater signal-to-noise than the standard hyperelastic approach. Riemannian priors regularize the full multivariate deformation tensor, yielding statistics on deformations that are unbiased in the associated Log-Euclidean metrics. Compared with standard continuum-mechanical registration, these Riemannian fluid models may more sensitively detect disease effects on the brain. 1

Abstract. We deal with the situation that a large number of hypotheses is investigated and sampling costs are constrained. Instead of distributing the sample size over the hypotheses in a single-stage design, a two stage design is considered: The first stage is used to screen the ”promising ” hypotheses which are then further investigated at the second stage. A multiple one-sided test procedure is proposed which aims at the control of the false discovery rate [1]. It is based on individual p-values appropriately defined for the two-stage design and explicitly worked out for the case of independent normally distributed test statistics with known variances. Asymptotically optimal designs are derived depending on the number of null hypotheses, the total costs, the proportion of true null hypotheses and a common effect size under the alternative. It can be shown, that the power of the two-stage design is impressively larger than the power of the corresponding single-stage design with equal costs. Extensions for the case of unknown variances, distributed effect sizes under the alternatives, correlated test statistics and the two-sided test are investigated by simulations.

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Abstract. The problem of multiple testing is rarely addressed in disease mapping or descriptive epidemiology. This issue is relevant when a large number of small areas or diseases are analysed. Control of the family wise error rate (FWER), for example via the Bonferroni correction, is avoided because it leads to loss of statistical power. To overcome such difficulties, control of the false discovery rate (FDR), the expected proportion of false rejections among all rejected hypotheses, was proposed in the context of clinical trials and genomic data analysis. FDR has a Bayesian interpretation and it is the basis of the so called q-value, the Bayesian counterpart of the p-value. In the present work, we address the multiplicity problem in disease mapping and show the performance of the FDR approach with two real examples and a small simulation study. The examples consider testing multiple diseases for a given area or multiple areas for a given disease. Using unadjusted p-values for multiple testing, an inappropriately large number of areas or diseases at altered risk are identified, whilst FDR procedures are appropriate and more powerful than the control of the FWER with the Bonferroni correction. We conclude that the FDR approach is adequate to screen for high/low risk areas or for disease excess/deficit and useful as a complementary procedure to point estimates and confidence intervals.